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Last time: Theorem 21 (Groves–Manning–Osin): If is hyperbolic rel then there exists a finite subset such that if then
(a) is injective;
(b) is hyperbolic rel .

Theorem 22 (Gromov, Olshanshkii, Delzant): If is hyperbolic relative to the infinite cyclic then there is a such that for all there exists a hyperbolic such that for each .

The proof is an easy application of Groves–Manning–Osin.

Definition: If (infinite cyclic) is malnormal then we say are independent. A group G is omnipotent if for every independent there exists a such that for all there exists a homomorphism $\phi$ from to a finite group such that for all .

Omnipotence strengthens residual finiteness for torsionfree groups.

Exercise 29: If every hyperbolic group is residually finite then every hyperbolic group is omnipotent.

We’ll finish off by talking about a similar theorem of Agol–Groves–Manning. I’m going to seem a little cavalier about torsion. This is OK. In fact, if every hyperbolic group is residually finite then every hyperbolic group is virtually torsionfree.

Theorem 22 (Agol–Groves–Manning): If every hyperbolic group is residually finite then every quasi-convex subgroup of any hyperbolic group is separable.

Let . The idea is to Dehn fill to get a new hyperbolic group in which the image is finite and . If we could do this, we would be done by residual finiteness. This works if is malnormal. But it probably isn’t. Fortunately, we can quantify how far is from being malnormal:

Definition: The height of is the maximal such that there are distinct cosets such that the intersection
is infinite.

H is height iff is finite. In a torsionfree group, is height iff is malnormal.

Theorem 23 (Gitik, Mitra, Rips, Sageev): A quasiconvex subgroup of a hyperbolic group has finite height.

Agol, Groves and Manning are able to prove:

Theorem 24: Let be a (torsionfree) residually finite hyperbolic group, and a quasiconvex subgroup of height . Let . Then is an epimorphism to a hyperbolic group such that
(i) is quasiconvex in ;
(ii) ;
(iii) has height .

The idea of the proof of Theorem 24 is to Dehn fill a finite index subgroup of a maximal infinite intersection of conjugates of . Theorem 22 is an easy consequence.

We still need to convince ourselves of some basic facts about the previous lecture, for example is the map injective?

Example: Cut the sphere along the equator. Then the diagram we have is

Definition: If or we say that G splits over C, and we call C the edge group. If or and is not or in the latter case, then we say that splits non-trivially.

Definition: Let be a connected graph (i.e. a 1-dimensional CW-complex). For each vertex (resp. edge ) let (resp. ) be a group. If are vertices adjoining an edge e then let be an injective homomorphism. This data determines a graph of groups.

We say that has:

underlying graph

vertex groups

edge groups

edge maps

Similarly, we have:

Definition: Let be a connected graph. For each vertex (resp. ) let (resp. ) be a connected CW-complex. If adjoin let be -injective continuous maps. This data determines a graph of spaces. It has underlying graph , vertex spaces , edge spaces , etc. The graph of spaces determines a space as follows: define

where for . We say that is a graph-of-spacesstructure (or decomposition) for .

Remark: There is a natural map (by collapsing all the edge and vertex spaces).

Given any graph of groups we can construct a graph of spaces with underlying graph by assigning and realizing the edge maps as continuous maps . We write for . This is well-defined up to homotopy equivalence.

Definition: The fundamental group of is just

Examples:

If then

If then

Let be an embedded multicurve (disjoint union of circles) inside a surface. Cutting along decomposes into a graph of spaces and into a graph of groups.

Note: The edge maps of are only defined up to free (i.e. unbased) homotopy. Translated to , this means that only the conjugacy class of in matters.

Remark: The map induces a surjection

Here’s a way to construct a graph of groups. Let’s suppose acts on a tree without edge inversions (we can do this by subdividing edges if necessary). Let . The group acts diagonally on The quotient has a structure of a graph of spaces. The underlying graph is and there is a natural map .

Let be a vertex below . The preimage of is just where is the stabilizer of . Similarly, for below , the preimage of is

If adjoins then so the edge map is a covering map and therefore -injective. We have defined a graph of spaces and since is simply connected.

Applying to everything, we have a graph of groups . Its underlying graph is . Its vertex groups are the vertex stabilizers of , its edge groups are the edge stabilizers, and the edge maps are the inclusions. Also, .